Annals of Mathematics, 150 (1999), 267–282
Nonsymmetric Koornwinder polynomials
and duality
By Siddhartha Sahi*
1. Introduction
In the fundamental work of Lusztig [L] on affine Hecke algebras, a special
en . The affine Hecke algebra is a
role is played by the root system of type C
deformation of the group algebra of an affine Weyl group which usually depends
on as many parameters as there are distinct root lengths, i.e. one or two for
en case, the Hecke algebra H has
an irreducible root system. However in the C
three parameters, corresponding to the fact that there is a simple coroot which
is divisible by 2.
Recently, Cherednik [C1]–[C3] has introduced the notion of a double affine
Hecke algebra, and has used it to prove several conjectures on Macdonald
polynomials. These polynomials, and Cherednik’s double affine Hecke algebra,
involve two or three parameters, i.e., one more than the number of root lengths.
In this paper, motivated by the work of Noumi [N], we define a double
en case, which depends on three additional paaffine Hecke algebra for the C
rameters, making six altogether. The associated orthogonal polynomials are
precisely the six-parameter family of polynomials Pλ introduced by Koornwinder in [Ko].
These polynomials are themselves quite remarkable. Every symmetric
Macdonald polynomial [M] associated to a classical root system (i.e. those
of types A, B, C, D, and the two classes of type BC, not considered by
Cherednik) can be obtained from the Pλ by a suitable limiting procedure [D].
Moreover, for n = 1, the Pλ become the Askey-Wilson polynomials, which sit
atop an impressive hierarchy of orthogonal polynomials in one variable [AW].
Koornwinder and Macdonald have formulated several conjectures for these
polynomials, which are analogous to those proved by Cherednik for Macdonald polynomials. These are the “constant term,” “norm,” “evaluation,” and
*This work was supported by an NSF grant, and carried out in part during the author’s visit
to Japan. The author wishes to thank M. Wakayama, H. Ochiai, K. Mimachi for their hospitality,
and M. Noumi for explaining the results of [N].
268
SIDDHARTHA SAHI
“duality” conjectures. For a certain five-parameter subfamily of the Pλ , these
conjectures were proved by van Diejen in [D].
In the general (six-parameter) setting, van Diejen has shown that either
of the last two conjectures implies the other three. One of the results in this
paper is a proof of the duality conjecture which implies all the rest by van
Diejen’s work.
Here is an outline of the paper: After a brief summary of the relevant
results of Koornwinder and Noumi, we define the six-parameter double affine
Hecke algebra H and establish its basic properties, including the existence of
an involution. Next, we introduce certain commutators Si in H, called the
intertwiners, and use them to construct a family of polynomials {Eα }. We
call these the nonsymmetric Koornwinder polynomials, and we describe their
relationship to the Pλ . Finally, we establish the duality conjecture for Pλ
together with its analog for Eα .
A substantial part of this paper is directly motivated by the results of
Cherednik in the two-/three-parameter setting. The idea of using intertwiners
as creation operators was introduced in [K], [KS] and [S] for GLn , and in [C4]
for other root systems.
We have avoided one layer of notational complexity by identifying the
coroot lattice of Cn with Zn . Thus we have suppressed explicit reference to
roots and weights. Implicitly, though, these are ubiquitous.
We have also obtained fairly precise results concerning the orthogonality
and triangularity of the nonsymmetric Koornwinder polynomials, which we
shall report elsewhere.
Finally, we remark that according to the note added in proof to [D], Macdonald has informed van Diejen that he has proved the evaluation conjecture.
By van Diejen’s work, this would also imply the duality conjecture for the Pλ ,
though not for the Eα .
2. Preliminaries
In this section we briefly recall some results of Koornwinder, Lusztig, and
Noumi which we shall need. For more details the reader should consult [Ko],
[L], and [N].
We fix six indeterminates q, t, t0 , tn , u0 , un , and let F be the field of rational
functions in their square roots. We also define
(1)
1/2 −1/2
1/2 −1/2
, c = q 1/2 t0 u0 , d = −q 1/2 t0 u0
a = tn1/2 u1/2
n , b = −tn un
1/2 1/2
.
Let R = F
be the ring of Laurent polynomials in n variables
over the field F, and let S be the subring consisting of symmetric polynomials, i.e. those which are invariant under permutations and inversions of the
variables.
±1
[x±1
1 , · · · , xn ]
NONSYMMETRIC KOORNWINDER POLYNOMIALS
269
2.1. Koornwinder polynomials. In [Ko], Koornwinder defined a basis {Pλ }
of S which is indexed by λ ∈ Zn with λ1 ≥ · · · ≥ λn ≥ 0, and defined as follows:
±1
Let Tq,xi denote the ith q-shift operator acting on R := F[x±1
1 , · · · , xn ] by
Tq,xi f (x1 , · · · , xi , · · · , xn ) := f (x1 , · · · , qxi , · · · , xn ).
Consider the following q-difference operator
D :=
n
X
Φi (x)(Tq,xi − 1) +
i=1
n
X
−1
Φi (x−1 )(Tq,x
− 1)
i
i=1
where Φi (x) is a rational function in x1 , · · · , xn defined by
³
´
−1
n
1
−
tx
(1 − txi xj )
x
Y
i
j
(1 − axi ) (1 − bxi ) (1 − cxi ) (1 − dxi )
¡
¢
¡
¢
³
´
.
Φi (x) :=
1 − x2i 1 − qx2i
1 − xi x−1 (1 − xi xj )
j=1
j6=i
j
Koornwinder showed that D preserves S and is diagonalizable with distinct eigenvalues
n h
i
X
q −1 abcdt2n−i−1 (q λi − 1) + ti−1 (q −λi − 1) .
dλ =
i=1
The Koornwinder polynomial Pλ is characterized by the equation
(2)
DPλ = dλ Pλ ,
together with the condition that the coefficient of xλ := xλ1 1 · · · xλnn in Pλ is 1.
It turns out that Koornwinder’s operator D is one among a commuting
family of difference operators, all of which are simultaneously diagonalized by
the Pλ . These higher operators were constructed abstractly by Noumi, and
explicitly by van Diejen.
To describe the results of Noumi, we need to introduce some additional
notation.
en is
2.2. The affine Weyl group. The affine Weyl group W of type C
generated by elements s0 , s1 , · · · , sn which satisfy s2i = 1 and, for n > 1, also
satisfy the braid relations
si sj si · · · = sj si sj · · ·
with two, three, or four terms on each side accordingly as i and j are connected
by zero, one or two lines in the Coexeter graph
0 == 1 −−2 −− · · · · · · · · · −−(n − 2) −−(n − 1) == n.
The (finite) Weyl group W0 of type Cn is the subgroup generated by s1 , · · · , sn .
W has a natural faithful affine action on V = Rn in which
(3)
sn · v = (v1 , · · · , vn−1 , −vn ),
s0 · v = (−v1 − 1, v2 , · · · , vn )
270
SIDDHARTHA SAHI
and the other si act by interchanging vi and vi+1 . Indeed W is the semidirect
product of W0 and τ (Zn ), where τ v denotes the translation by v. In terms of
the generators,
(4)
τ v = τ1v1 · · · τnvn ,
−1
τi = (si · · · sn−1 )(sn · · · s0 )(s−1
1 · · · si−1 ).
Since the W -action on V is affine, we get a representation of W on the
space Ve of affine linear functionals on V , which we identify with V × R δ as
follows:
(5) hv + rδ, v 0 i := v10 v1 + · · · + vn0 vn + r;
hw(v + rδ), v 0 i := hv + rδ, w−1 · v 0 i.
This representation is given explicitly by
(6)
i 6= 0
si (v + rδ) = si v + rδ,
s0 (v + rδ) = (−v1 , v2 , · · · , vn ) + (r − v1 )δ.
We define an exponential map from the lattice
(7)
xv+kδ := q −k xv11 · · · xvnn ,
Zn × Zδ ⊂ Ve to R by
v ∈ Zn , k ∈ Z.
Since the lattice is preserved under (6), we get a representation of W on R by
putting
(8)
v
v)
) := xw(e
;
w(xe
ve ∈ Zn × Zδ.
Then W acts by algebra homomorphisms and S = RW0 . Explicitly we have
(9)
s0 f (x) = f (qx−1
1 , x2 , · · · , xn )
si f (x) = f (x1 , · · · , xi+1 , xi , · · · , xn )
sn f (x) =
i 6= 0, n
f (x1 , · · · , xn−1 , x−1
n )
τi = Tq,xi .
en
2.3. The affine Hecke algebra. The affine Hecke algebra H of type C
is generated over F by elements T0 , T1 , · · · , Tn which satisfy the same braid
relations as the si , and also satisfy
Ti − Ti−1 = ti
1/2
−1/2
− ti
where t1 = · · · = tn−1 = t.
The elements T1 , · · · , Tn generate the (finite) Hecke algebra H0 of type Cn .
H and H0 have natural bases {Tw } consisting of w in W and W0 , respectively,
where
(10)
Tw = Ti1 · · · Til
if w = si1 · · · sil is a reduced (i.e. shortest) expression of w in terms of the si .
NONSYMMETRIC KOORNWINDER POLYNOMIALS
271
The analogs of the translations τi in (4) are the elements
(11)
−1
Yi = (Ti · · · Tn−1 )(Tn · · · T0 )(T1−1 · · · Ti−1
),
i = 1, · · · , n.
Lusztig [L] has shown that the Yi commute pairwise and generate a subalgebra
RY ≈ F[Y1±1 , · · · , Yn±1 ],
and that multiplication gives us a vector space isomorphism
H0 ⊗ RY ≈ H.
(12)
2.4. The Noumi representation. Let si act on R by (9); then Noumi in
[N] has shown that the following map π extends to a representation of H
on R:
(13)
±1/2
π(T0±1 ) := t0
±1/2
π(Ti±1 ) := ti
−1/2 (1
+ t0
−1/2 (1
−1
− cx−1
1 )(1 − dx1 )
(s0 − 1)
1 − qx−2
1
− ti xi x−1
i+1 )
(si − 1) i 6= 0, n
(1 − xi x−1
i+1 )
(1 − axn )(1 − bxn )
π(Tn±1 ) := tn±1/2 + t−1/2
(sn − 1),
n
1 − x2n
+ ti
where a, b, c, d are as in (1).
Moreover if f is in S then π(f ) := π (f (Y1 , · · · , Yn )) preserves S. Noumi
showed that the restriction of π(Y1 + · · · + Yn + Y1−1 + · · · + Yn−1 ) to S is a linear
combination of the Koornwinder operator D and a scalar. This means that the
Koornwinder polynomials are simultaneous eigenfunctions for the π(f ). More
precisely, Pλ is characterized by
(14)
π(f )Pλ (x) = f (q λ+ρ )Pλ (x),
where q λ+ρ means (q λ1 +ρ1 , · · · , q λn +ρn ) and ρ is defined by
p
(15)
q ρi = stn−i , with s := (t0 tn )1/2 = q −1 abcd.
Remark. The fact that π extends to a representation can be derived from
Proposition 3.6 in [L], along the lines suggested in Proposition 4.6 of [M].
3. The double affine Hecke algebra
We now introduce the algebra H which will be the principal object of
study in this paper. For convenience, we will write Z ∼ z as an abbreviation
for the relation
Z − Z −1 = z 1/2 − z −1/2 .
272
SIDDHARTHA SAHI
Definition. Let H be the algebra generated over F by elements Ti±1 ,
i = 0, · · · n, and commuting elements Xi±1 , i = 1, · · · , n, subject to the relations:
(i) Ti ∼ ti ,
en braid relations for the Ti ’s,
(ii) the C
(iii) Ti Xj = Xj Ti if |i − j| > 1, or if i = n and j = n − 1,
(iv) Ti Xi = Xi+1 Ti−1 , i = 1, · · · , n − 1,
(v) Xn−1 Tn−1 ∼ un ,
(vi) U0 ≡ q −1/2 T0−1 X1 ∼ u0 .
If we set u0 = un = 1 and t0 = tn , then H specializes to the threeparameter double affine Hecke algebra considered in [C1] for the affine root
en .
system C
Our definition is motivated by the following considerations:
Define a map π from the generators of H to End(R) by letting π(Ti±1 ) be
as in (13), and letting π(Xi±1 ) be the operator of multiplication by x±1
i .
3.1. Theorem.
The map π extends to a representation of H on R.
Proof. We need only verify that (i)–(vi) hold for π(Ti±1 ) and π(Xi±1 ).
The relations (i) and (ii) follow from Noumi’s result, and (iii) and (iv) are
easily verified using the formulas. For (v) we have
(1 − axn )(1 − bxn ) −1
(xn sn − x−1
n )
1 − x2n
−1/2 (1 − axn )(1 − bxn )
Tn X n →
7 t1/2
(sn xn − xn ).
n xn + tn
1 − x2n
−1/2
Xn−1 Tn−1 7→ t−1/2
x−1
n
n + tn
−1
−1
2
Since sn xn = x−1
n sn and xn − xn = −xn (1 − xn ), we get
1/2
−1/2 −1
Xn−1 Tn−1 − Tn Xn 7→ t−1/2
x−1
xn (1 − axn )(1 − bxn )
n
n − tn xn − tn
−1/2
= −(t1/2
ab)xn + tn−1/2 (a + b).
n + tn
1/2 1/2
1/2 −1/2
1/2
Substituting a = tn un and b = −tn un , this becomes un
proving (v).
Relation (vi) can be proved similarly using s0 x1 = qx−1
1 s0 .
3.2. Theorem.
−1/2
− un
The representation π is faithful.
Proof. We first note that in any word in H involving the generators, the
relations (i)–(vi) allow us to commute the Ti ’s past the Xj ’s. Thus every
element of H can be written as a linear combination of X α Tw , where Xα =
X1α1 · · · Xnαn and Tw is as in (10).
NONSYMMETRIC KOORNWINDER POLYNOMIALS
273
Suppose a nontrivial linear combination maps to 0 under π. We then get
X
cw,α xα π(Tw ) = 0
in End(R), where cw,α are scalars in F, not all zero.
The left side is a rational expression in the square roots of q, t, t0 , tn , u0 , un ,
and we consider what happens if we specialize the last five indeterminates to
1. By clearing denominators and eliminating common factors, we may ensure
that at least some of the cw,α have nonzero specializations; by (1), (10), and
(13), π(Tw ) specializes to the action of w as in (9).
Thus we get a nontrivial dependence relation in End(R) of the form
X
cw,α xα w = 0, w ∈ W, α ∈ Zn .
Since W = W0 τ (Zn ), we can rewrite this as
X
cw,α,β xα wτ β = 0, w ∈ W0 , α, β ∈ Zn .
Collecting the terms for β, we get
X
xα wpw,α (τ1 , · · · , τn ) = 0, w ∈ W0 , α ∈ Zn ,
P
where pw,α (x) is the Laurent polynomial β cw,α,β xβ , Since τi (xγ ) = (q γi )xγ ,
applying the expression to xγ we obtain
X
xα+wγ pw,α (q γ1 , · · · , q γn ) = 0.
It follows that pw,α (q γ1 , · · · , q γn ) = 0 for all γ in Zn outside the finite union
of hyperplanes determined by the conditions α + wγ = α0 + w0 γ for α, α0 , w, w0
occuring in the last expression. But then pw,α must be identically 0, and we
conclude that all cw,α,β = 0, contrary to the assumption.
Let us define RX := F[X1±1 , · · · , Xn±1 ]; then the above proof shows:
3.3. Corollary.
are injective.
The natural maps from H, H0 , RY and RX into H
We shall identify the above algebras with their images in H. Then we
have:
3.4. Corollary. The multiplication maps from RX ⊗ H and RX ⊗ H0
⊗ RY into H are linear isomorphisms.
We conclude this section by giving an intrinsic definition of the representation π. First, by the definition of H, it is clear that the map
(16)
1/2
χ : Ti 7→ ti ,
i = 0, · · · , n
extends to a one-dimensional character of H.
274
SIDDHARTHA SAHI
3.5. Proposition.
The representation π is isomorphic to IndH
H (χ).
Proof. The induced representation is H/I where I is the left ideal of H
1/2
generated by Ti − ti , i = 0, · · · , n. On the other hand, π ≈ H/J where J is
the annihilator of the cyclic vector 1 ∈ R. It remains to show that I = J .
1/2
First, since si (1) = 1 it follows from formula (7) that π(Ti − ti )(1) = 0,
and so
I ⊆ J.
1/2
Next consider the left ideal I in H generated by Ti − ti . Then we have
H = F + I; thus RX ⊗ H = RX ⊗ F + RX ⊗ I. Applying Corollary 3.4 we
conclude that
H = RX + I.
This mean that the isomorphism RX → R given by Xi 7→ xi = π(Xi ) · 1
can be factored as the following sequence of surjective maps
RX → H/I → H/J → R.
In particular, the middle map is an isomorphism, and so I = J .
4. The involution
Let ε denote the involution on F which sends q, t, tn , u0 to their inverses,
and which maps
t0 7→ u−1
n .
We shall show that ε extends to an involution on H. First we prove
4.1. Lemma.
Un ∼ un .
Let Un ≡ X1−1 T0 Y1−1 = X1−1 T1−1 · · · Tn−1 · · · T1−1 , then
−1
, and applying this repeatedly,
Proof. By (iv) we have Xi−1 Ti−1 = Ti Xi+1
we get
−1
Un = (T1 · · · Tn−1 )(Xn−1 Tn−1 )(Tn−1
· · · T1−1 ).
Thus Un is conjugate to Xn−1 Tn−1 and the lemma follows from relation (v).
4.2. Theorem. The map ε extends to an involution of H which maps
Xi to Yi , sends T1 , · · · , Tn to their inverses, and maps
T0 7→ Un−1 .
Proof. We first verify that the ε-transforms of (i)–(vi) hold in H. For
i 6= 0 the relation (i) becomes Ti−1 ∼ t−1
i , which is implied by Ti ∼ ti ; while
−1
−1
for i = 0 it becomes Un ∼ un , which follows from Lemma 4.1.
NONSYMMETRIC KOORNWINDER POLYNOMIALS
275
All the ε-transforms of the braid relations (ii) are immediate, except for
Un−1 T1−1 Un−1 T1−1 = T1−1 Un−1 T1−1 Un−1 .
?
We shall check this directly. Write Φ = T2 · · · Tn · · · T2 ; then X1 Φ = ΦX1 ,
and we get
T1−1 Un−1 T1−1 Un−1 = ΦT1 X1 ΦT1 X1 = ΦT1 ΦX1 T1 X1 = ΦT1 ΦX1 X2 T1−1
Un−1 T1−1 Un−1 T1−1 = T1 ΦT1 X1 ΦT1 X1 T1−1 = T1 ΦT1 ΦX1 T1 X1 T1−1
= T1 ΦT1 ΦT1−1 X1 X2 T1−1 .
By multiplying both sides by T1 X1−1 X2−1 T1 on the right, it suffices to show
?
(T2 · · · Tn · · · T2 )T1 (T2 · · · Tn · · · T2 )T1 = T1 (T2 · · · Tn · · · T2 )T1 (T2 · · · Tn · · · T2 ).
We apply T2 T1 T2 = T1 T2 T1 to both sides and commute the resulting T1 ’s
as far to the extremes as possible. Using T1 T2 T1 = T2 T1 T2 once on each side
and cancelling, we get
?
(T3 · · · Tn · · · T3 )T2 (T3 · · · Tn · · · T3 )T2 = T2 (T3 · · · Tn · · · T3 )T2 (T3 · · · Tn · · · T3 ).
Iterating, we reach the true relation Tn Tn−1 Tn Tn−1 = Tn−1 Tn Tn−1 Tn , which
proves (ii).
The ε-transforms of (iii)–(iv) are easily checked. For (v) we get
−1
)
ε(Xn−1 Tn−1 ) = Yn−1 Tn = (Tn−1 · · · T1 )T0−1 (T1−1 · · · Tn−1
which is conjugate to T0−1 . Hence the desired relation follows from T0 ∼ t0 .
Finally, for (vi) we have
ε(U0 ) = q 1/2 Un Y1 = q 1/2 (X1−1 T0 Y1−1 )Y1 = q 1/2 X1−1 T0 = U0−1 .
Thus (vi) follows from its original counterpart.
It follows that ε is a homomorphism, and it remains only to prove that
ε2 = 1. Since H is generated by {Ti , Xi , Yi | i = 1, · · · , n}, it suffices to show
that ε(Yi ) = Xi for all i. But
ε(Y1 ) = T1−1 · · · Tn−1 · · · T1−1 Un−1 = X1 .
Since Ti Xi Ti = Xi+1 and Ti−1 Yi Ti−1 = Yi+1 , the result follows for all i by
induction.
5. Intertwiners
In this section we introduce certain commutators in H, and prove that
they enjoy a crucial intertwining property with respect to the commutative
family RY .
276
SIDDHARTHA SAHI
Definition. We define operators Si in H as follows:
(17)
Si := [Ti , Yi ], i = 1, · · · , n;
S0 := [Y1 , Un ].
We also introduce the following notation, analogous to (7):
(18) X v+kδ := q −k X1v1 · · · Xnvn ;
5.1. Theorem.
(19)
Y v+kδ := q k Y1v1 · · · Ynvn ,
For all ve in
v ∈ Zn , k ∈ Z.
Zn × Zδ,
v
v)
Ye
Si = Si Y si (e
;
i = 0, · · · , n.
Proof. By Theorems 3.2 and 4.2, it is enough to prove the π ◦ ε transform
of (19). Applying π ◦ ε to (18) and (17), we get:
v
v
v)
v)
) = xe
; π ◦ ε(Y si (e
) = xsi (e
π ◦ ε(Y e
π ◦ ε(S0 ) = [x1 , π(T0−1 )]; π ◦ ε(Si ) = [π(Ti−1 ), xi ], i = 1, · · · , n.
Now, an easy calculation in End(R), using formulas (13) gives
 −1/2
−1
−1

 t0 x1 (1 − cx1 )(1 − dx1 ) i = 0
(20) π ◦ ε(Si ) = φi (x)si ; φi (x) = tn−1/2 x−1
i=n
n (1 − axn )(1 − bxn )

 −1/2
−1
ti
xi+1 (1 − ti xi xi+1 )
i 6= 0, n.
Thus the π◦ε transform of (19) becomes the following assertion in End(R):
?
v
v)
xe
φi (x)si = φi (x)si xsi (e
.
After cancelling the φi , this follows from (8).
5.2. Corollary. Let a0 , b0 , c0 , d0 be the ε transforms of a, b, c, d, then

−1
−1
 un q −1 (1 − c0 Y1 )(1 − d0 Y1 )(1 − qc0 Y1 )(1 − qd0 Y1 ) i = 0
2
(21) Si = tn (1 − a0 Yn )(1 − b0 Yn )(1 − a0 Yn−1 )(1 − b0 Yn−1 )
i=n

−1
−1
−1 −1
ti Yi Yi+1 (1 − ti Yi Yi+1 )(1 − ti Yi Yi+1 )
i 6= 0, n.
Proof. By (20), we get π ◦ ε(Si2 ) = φi si φi si = φi si (φi )s2i = φi si (φi ), and
the result follows from the explicit formula for φi .
In the next section we will use the Si ’s as creation operators for the Eα ,
starting with the constant function 1, which is an eigenfunction of Yi , satisfying
(22)
π(Yi )(1) = q ρi (1);
i = 1, · · · , n.
1/2
This is an immediate consequence of the equation π(Ti )(1) = ti (1) and the
definitions of Yi and ρ in (11) and (15). To describe the other eigenvalues, we
proceed as follows:
NONSYMMETRIC KOORNWINDER POLYNOMIALS
Definition. For α in
277
Zn , we define
wα := the shortest element in W0 such that wα−1 · α is a partition;
α := α + wα · ρ where ρ is as in (15);
Rα := the space of all f ∈ R satisfying Yi f = q αi +(wα ·ρ)i f for all i.
Alternatively, wα in W0 = (±1)n Sn can be described as wα := σα πα ,
where σα ∈ (±1)n is simply (sgn(α1 ), · · · , sgn(αn )) with sgn(0) defined to be
1; and πα is the permutation in Sn defined as follows: order the indices first
by decreasing |αi |, then for fixed |αi | from left to right for αi ≥ 0, and finally
from right to left for αi < 0.
For example if α = (−2, 2, 1, −1, 0, 1, −1), then σα = (−1, 1, 1, −1, 1, 1, −1)
and πα is the permutation (2, 1, 3, 6, 7, 4, 5).
5.3. Theorem.
Rα to Rsi ·α .
If si · α 6= α, then π(Si ) is a linear isomorphism from
Proof. Let f ∈ Rα . Then by (5) and (18) it follows that for all ve in
Zn × Zδ,
v
v ,α+wα ·ρi
Ye
(f ) = q he
f.
Let us write α = α + wα · ρ. Then from Theorem 5.1 and (3) we get
v
v)
v ),αi
v ,si ·αi
π(Y e
)π(Si )f = π(Si )π(Y si (e
)f = q hsi (e
π(Si )f = q he
π(Si )f.
Thus to prove that π(Si )f ∈ Rsi ·α , it suffices to show that si · α 6= α
implies
(23)
si · α = si · α;
i = 0, · · · , n.
For i = 0, write β = s0 · α = (−α1 − 1, α2 , · · · , αn ). Then we claim that
the permutations πβ and πα are the same. Indeed if α1 is positive then, in the
ordering corresponding to πα , the index 1 is the first among the indices j with
|αj | = α1 ; while in the ordering for πβ , 1 is the last index in the higher group of
indices satisfying |βj | = α1 + 1. Thus the relative position of 1 with respect to
other indices stays the same, as do those of other indices with respect to each
other. A similar argument works if α1 is negative, and taking into account the
sign change we conclude that:
β 1 = β1 + (wβ · ρ)1 = −α − 1 − (wα · ρ)1 = −α1 − 1;
β i = αi , i ≥ 2,
which is precisely the content of (23) for i = 0.
The argument for i ≥ 1 is similar and simpler. We observe that if β := si ·α
is different from α, then wβ equals si wα . Since si acts linearly, we get
si · α = si · α + si · (wα · ρ) = β + wβ · ρ = β = si · α.
278
SIDDHARTHA SAHI
Thus we conclude that π(Si ) maps Rα into Rsi ·α for all i ≥ 0. But, by
Corollary 5.2, Si2 is in RY ; hence π(Si2 ) acts by a scalar ci on Rα , which can
be readily computed by substituting Yi = αi in (21). In particular, we see that
if si · α 6= α then ci is not zero. Thus π(Si ) is a linear isomorphism from Rα
to Rsi ·α , with inverse c−1
i π(Si ).
6. Nonsymmetric Koornwinder polynomials
In this section we will define the nonsymmetric Koornwinder polynomials.
The crucial result is:
6.1. Theorem.
The spaces Rα are all one-dimensional.
Proof. We first prove that the spaces Rα are nonzero. For α = 0, wα is
the identity in W0 and so α = ρ. Thus by (22), the constant functions belong
to Rα for α = 0. For other α ∈ Zn we use Theorem 5.3 together with the fact
that the affine action of W on Zn is transitive.
P
Now let f = cβ ∈F cβ xβ be a nonzero function in Rα . Then f satisfies
π(Yi )f := q (α+wα ·ρ)i f ;
i = 1, · · · , n.
As in the proof of Theorem 3.2, we set t, t0 , tn , u0 , un equal to 1 in the expression. Then ρ specializes to the zero vector, and Yi specializes to π(τi ) = Tq,xi .
Clearing denominators and eliminating common factors, we may also assume
that the cβ have finite specializations, not all zero. Letting g 6= 0 denote the
specialization of f , we get
Tq,xi g = q αi g
which means that g is a nonzero multiple of xα .
In particular, the coefficient cα has a nonzero specialization and so must
be nonzero. The result follows, since if there were two linearly independent
functions in Rα , we could construct a nonzero f with cα = 0.
The proof of the theorem shows that a function f in Rα is uniquely determined by the knowledge of the coefficient of xα in f .
Definition. The nonsymmetric Koornwinder polynomial Eα is the unique
polynomial in Rα in which the coefficient of xα is 1.
6.2. Theorem.
The polynomials Eα form a basis for R over F.
Proof. Let us consider the degree filtration R(0) ⊆ R(1) ⊆ · · · ⊆ R, where
R(k) is spanned by all monomials xα with |α| := |α1 | + · · · + |αn | ≤ k. We
claim that for |α| ≤ k,
Rα ⊆ R(k) .
NONSYMMETRIC KOORNWINDER POLYNOMIALS
279
For α = 0, we use Theorem 6.1 to see that Rα consists precisely of the
constants, which lie in R(0) . For other α, we observe that by (13) the filtration
is invariant under Ti and Yi , while Un = X1−1 T0 Y1−1 raises degree by at most
one. Thus S1 , · · · , Sn preserve the filtration while S0 raises degree by at most
one. Now any α can be obtained from 0 by applying a sequence of si ’s in which
s0 occurs exactly |α| times. Applying the corresponding Si ’s to R0 , we deduce
the claim.
It follows that the set {Eα : |α| ≤ k} is contained in R(k) and has the
same cardinality as the monomial basis. Therefore it suffices to prove that the
Eα are linearly independent. For this we choose a polynomial f in R which
takes distinct values on the finite set {q α : |α| ≤ k}. Then the Eα belong to
distinct eigenspaces under the operator π(f (Y1 , · · · , Yn )) and hence are linearly
independent.
6.3. Corollary.
The representation π is irreducible.
P
Proof. Let V be a π(H)-invariant subspace of R, and suppose f = cα Eα
belongs to V, with some cβ 6= 0. Choose a function g in R such that g(q β ) =
1/cβ , and g(q α ) = 0 for all other α for which cα 6= 0. Then applying
π(g(Y1 , · · · , Yn )) to f we conclude that Eβ belongs to V. Now applying the
π(Si )’s we conclude that every Eα belongs to V.
Next we consider the restriction of π to H.
Definition. For each partition λ we write Rλ for the subspace of R
spanned by the {Eα : α ∈ W0 · λ}.
6.4. Corollary.
direct sum.
The Rλ are irreducible π(H)-modules, and R is their
Proof. For the irreducibility, we repeat the previous argument without
involving S0 . The second assertion follows from Theorem 6.2.
Finally we discuss the connection with the symmetric Koornwinder polynomials Pλ .
6.5. Corollary.
The symmetric Koornwinder polynomial Pλ can
be characterized as the unique W0 -invariant polynomial in Rλ which has the
coefficient of xλ equal to 1.
Proof. We note that if α ∈ W0 · λ, then α = α + wα · ρ = wα · (λ + ρ).
In particular, if f is in S, then f (α) = f (λ + ρ), and so Rλ is precisely the
f (λ + ρ)-eigenspace of π(f (Y1 , · · · , Yn )) for f ∈ S. The result now follows from
the characterization (14).
¡P
¢ P
2 −1
Definition. Define C ∈ H0 by C :=
w∈W0 χ(Tw )
w∈W0 χ(Tw )Tw .
280
SIDDHARTHA SAHI
6.6. Corollary.
π(C) is a projection from Rλ to FPλ .
Proof. First of all, an easy calculation as in Lemma 2.5 of [S] shows that
1/2
1/2
Ti C = ti C for i = 1, · · · , n; hence π(Ti )π(C)f = ti π(C)f for all f ∈ R. By
(13), this implies that π(C)f is W0 -invariant, and so must be a multiple of Pλ .
Moreover, for f ∈ S, π(Tw )f = χ(Tw )f ; hence π(C) acts by the identity
on S.
7. Duality
Let
inverses.
†
denote the involution on
F which sends q, t, t0 , tn , u0 , un to their
7.1. Proposition.
The map † extends to an anti -involution on H
satisfying
Ti† = Ti−1 , Xi† = Xi−1 , Yi† = Yi−1 .
Proof. For the proof we merely observe that each defining relation of H
is
† -invariant.
Definition. We define the duality anti-involution
h∗ = ε(h† ) = ε(h)† ,
On F,
∗
∗
on H by
h ∈ H.
simply switches t0 and un ; while on the generators,
Ti∗ = Ti , Xi∗ = Yi−1 , Yi∗ = Xi−1 , i = 1 · · · , n;
T0∗ = Un .
We also extend ∗ from F to an involution on R by defining x∗i = x−1
for
i
−1
all i. Observe that if f is in S, then f is invariant under xi 7→ xi , and so f ∗
is obtained just by switching t0 and un in the coefficients of f .
∗
Next, we define ρ∗ by the requirement that q ρ = (q ρ )∗ . Explicitly,
∗
q ρi = (q ρi )∗ = ((t0 tn )1/2 tn−i )∗ = (un tn )1/2 tn−i .
The duality conjecture of Macdonald can be stated as follows:
7.2. Conjecture.
For any two partitions λ and µ we see that
Pµ∗ (q λ+ρ )
Pλ (q µ+ρ )
=
.
Pλ (q ρ∗ )
Pµ∗ (q ρ )
∗
This is seen to be equivalent to the formulation in (4.4) of [D], after the
easy verification that our definition of duality (t0 ↔ un ), is the same as that
in (4.1) of [D].
To establish Conjecture 7.2 and its analog for Eα , we introduce the following:
281
NONSYMMETRIC KOORNWINDER POLYNOMIALS
Definition. Let S be the map from H to
∗
S(h) := Fh (q −ρ );
7.3. Theorem.
F defined by
where Fh = π(h)(1) ∈ R.
We have S(h∗ ) = S(h)∗ for all h ∈ H.
Proof. By linearity and Corollary 3.4 it is enough to prove this for h of
the form X α Tw Y β , with α, β ∈ Zn , w ∈ W0 . Then by (16) and (22),
Fh = q hβ,ρi χ(Tw )xα ;
(24)
∗
and S(h) = q hβ,ρi χ(Tw )q −hα,ρ i .
On the other hand h∗ = X −β Tw∗ Y −α , and so
(25)
∗
Fh∗ = q −hα,ρi χ(Tw∗ )x−β ;
and S(h∗ ) = q −hα,ρi χ(Tw∗ )q hβ,ρ i .
For w ∈ W0 , χ(Tw ) only involves t and tn ; and Tw∗ is simply obtained from
Tw by reversing its product expansion (10) in terms of T1 , · · · , Tn . Thus
χ(Tw )∗ = χ(Tw ) = χ(Tw∗ );
Since
and (25).
∗
w ∈ W0 .
interchanges ρ and ρ∗ , the result now follows by comparing (24)
We now define scalars Eαβ , Pλµ in
∗
Eαβ := Eα∗ (q β )Eβ (q −ρ );
7.4. Theorem.
F by
∗
Pλµ := Pλ∗ (q µ+ρ )Pµ (q −ρ ).
∗ =E
∗
We have Eαβ
βα and Pλµ = Pµλ .
Proof. For the first assertion we consider h := Eα∗ (Y )Eβ (X). Then by the
definition of ∗ , we get h∗ = Eβ∗ (Y )Eα (X). Now,
Fh := π(Eα∗ (Y ))Eβ (x) = Eα∗ (q β )Eα (x),
and so S(h) = Eαβ . Similarly S(h∗ ) = Eβα , and the result follows from Theorem 7.3.
The second assertion is proved similarly by considering h := Pλ∗ (Y )Pµ (X).
7.5. Corollary. Conjecture 7.2 is true.
we get
Proof. Since Pλ is invariant under xi 7→ x−1
i
∗
∗
Pµλ := Pµ∗ (q λ+ρ )Pλ (q −ρ ) = Pµ∗ (q λ+ρ )Pλ (q ρ ).
On the other hand,
³
´∗
∗
∗
∗
∗
Pλµ
:= Pλ∗ (q µ+ρ )Pµ (q −ρ ) = Pλ (q −µ−ρ )Pµ∗ (q ρ ) = Pλ (q µ+ρ )Pµ∗ (q ρ ).
Thus the result follows from Theorem 7.4.
282
SIDDHARTHA SAHI
Rutgers University, New Brunswick, NJ
E-mail address: sahi@math.rutgers.edu
References
[AW] R. Askey and J. Wilson, Some Basic Hypergeometric Orthogonal Polynomials that
Generalize Jacobi Polynomials, Mem. A.M.S. 54 (1985).
[C1] I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of
Math. 141 (1995), 191–216.
[C2]
, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119–145.
[C3]
, Nonsymmetric Macdonald polynomials, IMRN, no. 10 (1995), 483–515.
[C4]
, Intertwining operators of double affine Hecke algebras, Selecta Math. 3 (1997),
459–495.
[D]
J. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126
(1996), 319–339.
[K]
F. Knop, Integrality of two variable Kostka functions, J. Reine Angew. Math. 482
(1997), 177–189.
[KS] F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials,
Invent. Math. 128 (1997), 9–22.
[Ko] T. Koornwinder, Askey-Wilson polynomials for root systems of type BC, Contemp.
Math. 138 (1992), 189–204.
[L]
G. Lusztig, Affine Hecke algebras and their graded version, J. A.M.S. 2 (1989), 599–
635.
[M]
I. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque, No. 237
(1996), 189–207.
[N]
M. Noumi, Macdonald-Koornwinder polynomials and affine Hecke rings, Sūrikaisekikenkyūsho Kōkyūroku 919 (1995), 44–55 (in Japanese).
[S]
S. Sahi, Interpolation, integarlity, and a generalization of Macdonald’s polynomials,
IMRN, no 10 (1996), 457–471.
(Received August 13, 1997)